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G-Adaptivity: optimised graph-based mesh relocation for finite element methods
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Trains a diffusion-based graph neural network mesh deformer end-to-end through Firedrake adjoints to directly minimise FEM L2 solution error for online r-adaptive meshing.
Problem
Classical r-adaptive meshing (Monge-Ampère) relies on solving an expensive meshing PDE and on interpolation-error proxies for FEM error, while prior ML surrogates inherit these proxies and cannot beat classical error reduction.
Approach
A GNN "diffformer" (graph diffusion with learned row-stochastic attention adjacency, forward-Euler steps) relocates mesh points and is trained by backpropagating the true FEM L2 error (gradients via Firedrake adjoint) plus an equidistribution regularizer; the diffusion structure provably avoids mesh tangling.
Key results
- On Poisson square, G-Adapt achieves 21.01% error reduction vs 12.69% for MA, at 88 ms vs 3780 ms.
- On Burgers' square rollout, G-Adapt achieves 27.17% error reduction vs 25.78% for MA, while UM2N gives -11.24%.
- On Navier-Stokes cylinder, G-Adapt achieves 26.36% error reduction in 49 ms, versus 25.55% for UM2N-g and 1.34% for UM2N.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use G-Adapt instead of MA | Mesh relocation for Poisson · Poisson Square | Error reduction (%): 12.69 $\pm$ 0.00 -> 21.01 $\pm$ 0.33 | Helped |
| Use G-Adapt instead of UM2N-g | Mesh relocation for Poisson · Poisson Convex Polygon | Error reduction (%): 15.00 $\pm$ 0.13 -> 16.84 $\pm$ 0.10 | Helped |
| Use G-Adapt instead of MA | Mesh relocation for time-evolving Burgers' equation · Burgers' Square Rollout | Error reduction (%): 25.78 $\pm$ 0.00 -> 27.17 ${\pm}$ 0.34 | Helped |
| Train G-Adapt for the specified remeshing frequency | Mesh relocation for Burgers' equation · Burgers' Square 10 Steps | Error reduction (%): 12.28 $\pm$ 0.00 -> 21.66 $\pm$ 3.13 | Helped |
| Use G-Adapt instead of UM2N-g | Mesh relocation for Navier--Stokes · Navier--Stokes | Error reduction (%): 25.55 ± 0.81 -> 26.36±1.37 | No clear effect |
| Use G-Adapt instead of UM2N-g | Mesh relocation for Poisson · Poisson non-convex geometries | Error reduction (%): 1.92 $\pm$ 0.02 -> 7.97 $\pm$ 0.04; 0.69 $\pm$ 0.08 -> 8.88 $\pm$ 0.24; $-0.96 \pm 0.04$ -> 6.62 $\pm$ 0.09; $-0.92 \pm | Helped |
| Use G-Adapt instead of MA at larger mesh scales | Poisson mesh relocation · Poisson Square 60x60 | Error reduction (%): 11.94 ± 1.50 -> 27.47 ± 0.89 | Helped |
| Remove the equidistribution regularizer (ablation) | G-Adapt · Poisson Square | Error reduction (%): 23.99 -> 22.42 | Hurt |
| Vary the equidistribution regularizer weight from 0.5 to 8 (ablation) | G-Adapt · Poisson Square | Error reduction (%): 22.95; 23.99; 23.21; 22.14; 20.96 | Mixed |
| Vary diffusion timestep and number of diffusion steps (ablation) | G-Adapt · Poisson Square | Error reduction (%): 10.41; 16.60; 18.95; 15.85; 21.82; 22.93; 12.97; 14.52; 20.68; 15.71; 20.27; 20.85; 19.54; 19.11; 22.30; 19.94; 23.11; 2 | Mixed |
Surprising findings
- Pretrained UM2N models yield negative error reduction on Burgers' rollout, attributed to out-of-distribution trajectories.
- The direct-optimisation reference (DirectOpt) reaches only 27.40% on Poisson square, indicating MA meshes are not optimal in error terms but gains from direct optimisation are limited.
Limitations
- Out-of-distribution generalisation: ML-based meshing may degrade on solutions with different scales or features from training data.
- UM2N baseline was trained with limited tuning (1000 epochs), so comparisons may be sensitive to training choices.
- Training requires Firedrake adjoint computations and reference solutions on fine meshes, though inference does not.
Experimental setup
Datasets: randomly sampled Gaussian source terms and boundary conditions on square, polygonal, and non-convex domains Navier-Stokes cylinder flow snapshots Burgers' equation Gaussian initial conditions
Benchmarks: Poisson square Poisson convex polygon Burgers' square rollout Burgers' square 10 steps Navier-Stokes cylinder non-convex Poisson geometries 3D Poisson cube
Metrics: relative L2 error reduction (%) relocation time (ms) aspect ratio
Baselines: Monge-Ampère (MA) UM2N UM2N-G DirectOpt
Builds on: UM2N Monge-Ampère mesh adaptation graph attention network Firedrake adjoint (pyadjoint)
Closest papers by meaning
- AMBER: Adaptive Mesh Generation by Iterative Mesh Resolution Prediction (NeurIPS 2025 · similarity 0.87)
- Better Neural PDE Solvers Through Data-Free Mesh Movers (ICLR 2024 · similarity 0.86)
- UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss (NeurIPS 2025 · similarity 0.85)
- Towards Universal Mesh Movement Networks (NeurIPS 2024 · similarity 0.84)
- GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems (NeurIPS 2026 · similarity 0.84)
- PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations (ICLR 2025 · similarity 0.83)
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